The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a graph embedded in the surface.

In particular, there is a 1-1 correspondence between spin structures on a surface and Kasteleyn orientation on a surface graph with dimer configuration. The number of non-equivalent Kasteleyn orientations of a surface graph of genus $g$ is $2^{2g}$ and is equal to the number of non-equivalent spin structures on the surface. Kasteleyn operator can be naturally identified with a discrete version of the Dirac operator. And the partition function of the dimer model is equal to the sum of $2^{2g}$ Pfaffians, reminiscent of the partition function of free fermions on a Riemann surface of genus $g$, which is a linear combination of $2^{2g}$ Pfaffians of Dirac operators.

My question is, given the above combinatorial description of spin structure, is there a way to write a local combinatorial description for the spin cobordism invariants in 2d, e.g. the Arf invariant? (see Wikipedia http://en.wikipedia.org/wiki/Arf_invariant for a definition of the Arf invariant, in particular, the section "The Arf invariant in topology".)


Here is a partial answer: Dimer Models (as a special case of the Ising Model) are an example of complex structures on surfaces. If you arrange an octagon and identify the opposing sides, the surface has genus 2 (with two special points) and then overlay a grid and we can count the number of dimer tilings. And that will depend on the arf invariant of the surface.

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You can find in the literature equivalences between dimer model and six-vertex, XXZ spin chain, Ising model, free fermions, Dirac operators etc. And there will always be a few details missing.

Here in this work of Julien Dubedat we get how the Kasteleyn matrices are discretization of the Cauchy-Riemann operators. So we are only getting the complex structure

Unfortunately that guy is interested in Stochastic Loewner Evolution (SLE) and there is hardly any topology in that discussion. Instead he will talk about Sobolev spaces and convergence to the Gaussian Free Field. I'd be more interested in the case of rectangular grids, but the tools are all there.

Going back to Reshetikhin and Cimasoni, the Arf invariant has to do with the Kasteleyn orientation in other words, Kasteleyn's formula says the number of tilings is the determinant of someting, but determinants have a sign and how to you make sure there is the correct sign? So the answer involves quadratic forms and the Arf invariant.

Ultimately he will prove there is an entire TQFT structure on dimer tilings. Maybe someone looked into that more carefully.


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